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1	Normal forms around lower dimensional tori of hamiltonian systems Villanueva Castelltort, Jordi 10 March 1997 (has links) L'objectiu bàsic d'aquesta tesi és l'estudi de la dinàmica a l'entorn de tors de dimensió baixa de sistemes hamiltonians analítics. Per aquest estudi l'eina fonamental és l'ús de formes normals al voltant d'aquests objectes.La formulació dels resultats d'aquesta memòria s'ha fet de manera adient per a la seva aplicació a models de mecànica celeste del món real. Per aquest motiu els resultats no es redueixen només al cas autònom, sinó que algun d'ells pren en consideració la possiblitat que les diferents perturbacions pugin dependre del temps de forma periòdica o quasiperiòdica. Aquests resultats s'apliquen per descriure la dinàmica d'alguns problemes d'interes per la Astronàutica. Per tant, els resultats obtinguts inclouen també aplicacions numèriques.Els resultats assolits en cadascun del capítols de la memòria es poden sintetitzar de la forma següent:Capítol 1.- Estudi de la dinàmica entorn d'un tor parcialment el.líptic d'un sistema Hamiltonià autònom. Es donen cotes inferiors pel temps de difusió entorn d'un tor totalment el.líptic, així com estimacions, en el cas general, de la densitat de tors invariants (de qualsevol dimensió) al voltant del tor inicial. Les estimacions en la velocitat de difusió i en la proximitat a 1 d'aquesta densitat, són exponencialment petites respecte la distància al tor inicial.Capítol 2.- Computació numèrica de formes normals al voltant d'òrbites periòdiques. Es desenvolupa un mètode per a calcular formes normals al voltant d'òrbites periòdiques el.líptiques de sistemes hamiltonians. Aquesta metodologia és aplicada numèricament a una òrbita periòdica del Problema Restringit de tres Cossos espaial. Els resultats d'aquest capítol es poden veure com una implementació numèrica del Capítol 1.Capítol 3.- Persistència de tors de dimensió baixa sota perturbacions quasiperiòdiques. Es mostra que un tor de dimensió baixa d'un sistema hamiltonià sotmès a una perturbació quasiperiòdica és pot continuar respecte el paràmetre perturbatiu, tot afegint a les freqüències bàsiques inicials les de la perturbació, excepte per un conjunt de mesura petita pel paràmetre. Al igual que en el Capítol 1 també s'estima la densitat de tors en el problema perturbat. En ambdós casos, les cotes obtingudes per la mesura dels tors pels qual no és possible provar existència són de tipus exponencialment petit.Apèndix. Es presenta un resultat obtingut de forma conjunta amb Rafael Ramírez-Ros sobre la reducció a coeficients constants de sistemes d'equacions lineals autònoms perturbats quasiperiòdicament. Es mostra que tal reducció és possible excepte un reste exponencialment petit en el tamany de la perturbació. effective stability invariant tori normal forms 1200. Matemàtiques 51 517
2	Numerical Methods for the Continuation of Invariant Tori Rasmussen, Bryan Michael 24 November 2003 (has links) This thesis is concerned with numerical techniques for resolving and continuing closed, compact invariant manifolds in parameter-dependent dynamical systems with specific emphasis on invariant tori under flows. In the first part, we review several numerical methods of continuing invariant tori and concentrate on one choice called the ``orthogonality condition'. We show that the orthogonality condition is equivalent to another condition on the smooth level and show that they both descend from the same geometrical relationship. Then we show that for hyperbolic, periodic orbits in the plane, the linearization of the orthogonality condition yields a scalar system whose characteristic multiplier is the same as the non-unity multiplier of the orbit. In the second part, we demonstrate that one class of discretizations of the orthogonality condition for periodic orbits represents a natural extension of collocation. Using this viewpoint, we give sufficient conditions for convergence of a periodic orbit. The stability argument does not extend to higher-dimensional tori, however, and we prove that the method is unconditionally unstable for some common types of two-tori embedded in R^3 with even numbers of points in both angular directions. In the third part, we develop several numerical examples and demonstrate that the convergence properties of the method and discretization can be quite complicated. In the fourth and final part, we extend the method to the general case of p-tori in R^n in a different way from previous implementations and solve the continuation problem for a three-torus embedded in R^8. Invariant tori Numerical analysis Invariant manifolds Flows Numerical analysis Invariant manifolds Mathematical models
3	Chaotic Scattering in Rydberg Atoms, Trapping in Molecules Paskauskas, Rytis 20 November 2007 (has links) We investigate chaotic ionization of highly excited hydrogen atom in crossed electric and magnetic fields (Rydberg atom) and intra-molecular relaxation in planar carbonyl sulfide (OCS) molecule. The underlying theoretical framework of our studies is dynamical systems theory and periodic orbit theory. These theories offer formulae to compute expectation values of observables in chaotic systems with best accuracy available in given circumstances, however they require to have a good control and reliable numerical tools to compute unstable periodic orbits. We have developed such methods of computation and partitioning of the phase space of hydrogen atom in crossed at right angles electric and magnetic fields, represented by a two degree of freedom (dof) Hamiltonian system. We discuss extensions to a 3-dof setting by developing the methodology to compute unstable invariant tori, and applying it to the planar OCS, represented by a 3-dof Hamiltonian. We find such tori important in explaining anomalous relaxation rates in chemical reactions. Their potential application in Transition State Theory is discussed. Periodic orbit theory Chaotic dynamics Hamiltonian systems Dynamical systems Symbolic dynamics Invariant tori Scattering (Physics) Rydberg states Differentiable dynamical systems Combinatorial dynamics Chaotic behavior in systems
4	Bifurcations of families of 1-tori in 4D symplectic maps Onken, Franziska 14 August 2015 (has links) (PDF) The dynamics of Hamiltonian systems (e.g. planetary motion, electron dynamics in nano-structures, molecular dynamics) can be investigated by symplectic maps. While a lot of work has been done for 2D maps, much less is known for higher dimensions. For a generic 4D map regular 2D-tori are organized around a skeleton of families of elliptic 1D-tori, which can be visualized by 3D phase-space slices. An analysis of the different bifurcations of the families of 1D-tori in phase space and in frequency space by computing the involved hyperbolic and elliptic 1D-tori is presented. Applying known results of normal form analysis, both the local and the global structure can be understood: Close to a bifurcation of a 1D-torus, the phase-space structures are surprisingly similar to bifurcations of periodic orbits in 2D maps. Far away the phase-space structures can be explained by remnants of broken resonant 2D-tori. / Die Dynamik Hamilton'scher Syteme (z.B. Planetenbewegung, Elektronenbewegung in Nanostrukturen, Moleküldynamik) kann mit Hilfe symplektischer Abbildungen untersucht werden. Bezüglich 2D Abbildungen wurde bereits umfassende Forschungsarbeit geleistet, doch für Systeme höherer Dimension ist noch vieles unverstanden. In einer generischen 4D Abbildung sind reguläre 2D-Tori um ein Skelett aus Familien von elliptischen 1D-Tori organisiert, was in 3D Phasenraumschnitten visualisiert werden kann. Durch die Berechnung der beteiligten hyperbolischen und elliptischen 1D-Tori werden die verschiedenen Bifurkationen der Familien von 1D-Tori im Phasenraum und im Frequenzraum analysiert. Die Anwendung bekannter Ergebnisse aus Normalformanalysen ermöglicht das Verständnis sowohl des lokalen, als auch des globalen Regimes. Nahe an der Bifurkation eines 1D-Torus sind die Phasenraumstrukturen denen von Bifurkationen periodischer Orbits in 2D Abbildungen überraschend ähnlich. Weit entfernt können die Phasenraumstrukturen als Überreste eines zerplatzten resonanten 2D-Torus erklärt werden. Hamilton'sche Systeme höherdimensionale Systeme Bifurkationen niederdimensionale invariante Tori Hamiltonian Systems higher-dimensional Systems Bifurkationen lower-dimensional invariant tori ddc:530 rvk:UO 4020
5	Bifurcations of families of 1-tori in 4D symplectic maps Onken, Franziska 14 August 2015 (has links) The dynamics of Hamiltonian systems (e.g. planetary motion, electron dynamics in nano-structures, molecular dynamics) can be investigated by symplectic maps. While a lot of work has been done for 2D maps, much less is known for higher dimensions. For a generic 4D map regular 2D-tori are organized around a skeleton of families of elliptic 1D-tori, which can be visualized by 3D phase-space slices. An analysis of the different bifurcations of the families of 1D-tori in phase space and in frequency space by computing the involved hyperbolic and elliptic 1D-tori is presented. Applying known results of normal form analysis, both the local and the global structure can be understood: Close to a bifurcation of a 1D-torus, the phase-space structures are surprisingly similar to bifurcations of periodic orbits in 2D maps. Far away the phase-space structures can be explained by remnants of broken resonant 2D-tori. / Die Dynamik Hamilton'scher Syteme (z.B. Planetenbewegung, Elektronenbewegung in Nanostrukturen, Moleküldynamik) kann mit Hilfe symplektischer Abbildungen untersucht werden. Bezüglich 2D Abbildungen wurde bereits umfassende Forschungsarbeit geleistet, doch für Systeme höherer Dimension ist noch vieles unverstanden. In einer generischen 4D Abbildung sind reguläre 2D-Tori um ein Skelett aus Familien von elliptischen 1D-Tori organisiert, was in 3D Phasenraumschnitten visualisiert werden kann. Durch die Berechnung der beteiligten hyperbolischen und elliptischen 1D-Tori werden die verschiedenen Bifurkationen der Familien von 1D-Tori im Phasenraum und im Frequenzraum analysiert. Die Anwendung bekannter Ergebnisse aus Normalformanalysen ermöglicht das Verständnis sowohl des lokalen, als auch des globalen Regimes. Nahe an der Bifurkation eines 1D-Torus sind die Phasenraumstrukturen denen von Bifurkationen periodischer Orbits in 2D Abbildungen überraschend ähnlich. Weit entfernt können die Phasenraumstrukturen als Überreste eines zerplatzten resonanten 2D-Torus erklärt werden. info:eu-repo/classification/ddc/530 ddc:530
6	Characterization of Quasi-Periodic Orbits for Applications in the Sun-Earth and Earth-Moon Systems Brian P. McCarthy (5930747) 17 January 2019 (has links) <div>As destinations of missions in both human and robotic spaceﬂight become more exotic, a foundational understanding the dynamical structures in the gravitational environments enable more informed mission trajectory designs. One particular type of structure, quasi-periodic orbits, are examined in this investigation. Speciﬁcally, eﬃcient computation of quasi-periodic orbits and leveraging quasi-periodic orbits as trajectory design alternatives in the Earth-Moon and Sun-Earth systems. First, periodic orbits and their associated center manifold are discussed to provide the background for the existence of quasi-periodic motion on n-dimensional invariant tori, where n corresponds to the number of fundamental frequencies that deﬁne the motion. Single and multiple shooting diﬀerential corrections strategies are summarized to compute families 2-dimensional tori in the Circular Restricted Three-Body Problem (CR3BP) using a stroboscopic mapping technique, originally developed by Howell and Olikara. Three types of quasi-periodic orbit families are presented: constant energy, constant frequency ratio, and constant mapping time families. Stability of quasi-periodic orbits is summarized and characterized with a single stability index quantity. For unstable quasi-periodic orbits, hyperbolic manifolds are computed from the diﬀerential of a discretized invariant curve. The use of quasi-periodic orbits is also demonstrated for destination orbits and transfer trajectories. Quasi-DROs are examined in the CR3BP and the Sun-Earth-Moon ephemeris model to achieve constant line of sight with Earth and avoid lunar eclipsing by exploiting orbital resonance. Arcs from quasi-periodic orbits are leveraged to provide an initial guess for transfer trajectory design between a planar Lyapunov orbit and an unstable halo orbit in the Earth-Moon system. Additionally, quasi-periodic trajectory arcs are exploited for transfer trajectory initial guesses between nearly stable periodic orbits in the Earth-Moon system. Lastly, stable hyperbolic manifolds from a Sun-Earth L<sub>1</sub> quasi-vertical orbit are employed to design maneuver-free transfer from the LEO vicinity to a quasi-vertical orbit.</div> Aerospace Engineering aerospace engineering trajectory trajectory design astrodynamics three body three body mechanics circular restricted three body problem CR3BP quasi-periodic orbits periodic orbits orbit mechanics celestial mechanics dynamical systems dynamical systems theory dynamics applications orbits astronautics differential corrections invariance invariant circle invariant torus torus invariant tori stroboscopic mapping manifolds Sun-Earth system Earth-Moon system stability multiple shooting stroboscopic map eclipse avoidance vertical orbit halo orbit distant retrograde orbit quasi-halo quasi-vertical lissajous quasi-DRO DRO NRHO quasi-NRHO synodic resonance trajectory arcs mission design astromechanics spaceflight mechanics dynamical structures

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